1d:T7b3,# Probability mass function, cumulative distribution function, quantile function and random generation of the DSL The function computes the probability mass function, the cumulative distribution function, the quantile function of the DSL and implements random generation. ```r ddlaplace(x, p, q) pdlaplace(x, p, q) qdlaplace(prob, p, q) rdlaplace(n, p, q) ``` ## Arguments - `x`: vector of quantiles - `p`: the first parameter $p$ in $(0,1)$ of the SDL - `q`: the second parameter $q$ in $(0,1)$ of the SDL - `prob`: vector of probabilities - `n`: number of observations ## Details The pmf of the SDL is given by $P(X=x; p, q)=\frac{(1-p)(1-q)}{1-pq}p^x; x=0,1,2,3,\dots$ $P(X=x; p, q)=\frac{(1-p)(1-q)}{1-pq}q^{|x|}; x=0,-1,-2,-3,\dots$ whereas the cumulative distribution function is given by $F(x; p, q)=P(X\leq x)=\frac{(1-p)q^{-\lfloor x\rfloor}}{1-pq},x<0$ $F(x; p, q)=P(X\leq x)=1-\frac{(1-q)p^{\lfloor x\rfloor+1}}{1-pq},x\geq 0$ ## Returns `ddlaplace` returns the probability of `x`; `pdlaplace` returns the cumulate probability of `x`; `qdlaplace` returns the `prob`- quantile; `rdlaplace` returns a random sample of size `n` from DSL. ## References Tomasz J. Kozubowski, Seidu Inusah (2006) A skew Laplace distribution on integers, **Annals of the Institute of Statistical Mathematics**, 58: 555-571 ## Author(s) Alessandro Barbiero, Riccardo Inchingolo ## See Also `ddlaplace2` ## Examples ```r # pmf p<-0.7 q<-0.45 x<--10:10 prob<-ddlaplace(x, p, q) plot(x, prob, type="h") prob<-ddlaplace(x, q, p) # swap the parameters plot(x, prob, type="h") ddlaplace(-4:4, 1:9/10, 9:1/10) # letting p and q be vectors # cdf p<-0.2 q<-0.5 x<-c(-3, -1, pi) pdlaplace(x, p, q) # quantile function p<-0.8 q<-0.4 prob<-c(0.2,0.5,0.8) x<-qdlaplace(prob, p, q) x # check upper<-pdlaplace(x, p, q) upper lower<-pdlaplace(x-1, p, q) lower lower<=prob & prob<=upper # random generation n<-100 p<-0.3 q<-0.5 x<-rdlaplace(n, p, q) x t<-table(x) t plot(t) ```

ddlaplace function